On the numerical solution of Helmholtz-type PDEs using an adaptive complex collocation technique

Abstract

In this paper1 we consider the numerical solution of Helmholtz-type Partial Differential Equations in convex polygons, using a numerical technique based on the unified transform method. The key step of this approach is the utilization of the so-called global relation, which is an equation that couples the integral transforms of the given and the unknown boundary data, characterizing a Dirichlet-to-Neumann map. Solving the global relation in the complex k-plane, results in the determination of the missing boundary values, given a computational domain and prescribed boundary conditions. The considered numerical technique depends on the partitioning of the computational domain into a predetermined number of concentric polygons where the solution is required. Starting from the boundaries and proceeding towards the center of the domain, a spatial-marching procedure is used in order to compute the solution at each concentric polygon, using the Dirichlet and Neumann values computed via the global relations. The global relations are solved numerically using a collocation method in the complex k-plane, adapting the k-parameter at each spatial level. Numerical results indicating the applicability of the proposed method are provided, along with discussions concerning the implementation details.